Question

Qual o valor da soma das raízes da equação senx + cosx = raiz de 1/2 em [0, 2pi]

Answer 1

x ∈ [0, 2π]
senx + sen(π/2 - x) =√1/2
2senx( x+π/2-x)/2 cos(x-π/2+x)/2 = √2/2
2sen(π/4) cos(x - π/4) = √2/2
2.√2/2 . cos(x-π/4) = √2/2
2cos(x-π/4) = 1
cos(x-π/4) = 1/2
cos(x-π/4) = cosπ/3 => x - π4 = π/3 => x = 7π/12 ou
cos(x- π/4) = cos(5π/3) => x - π/4 = 5π/3 => x = 23π/12
S = 7π/12 + 23π/12
S = 30π/12
S = 5π/2